Solides - 6e

Pavé droit, cube et prisme droit

Exercice 1 : Reconnaître les sommets/faces/arêtes d'un solide

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 500, "height": 350}}, "data": {"Cylinder_1": {"position": {"x": 0.0, "y": -5.0, "z": 20.0}, "rotation": {"x": -1.57079632679, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "GenericRightCylinder", "path": "Triangle_0", "amount": 40}, "Triangle_0": {"type": "Path2D", "path": [{"moveTo": {"x": 0.0, "y": 0.0}}, {"lineTo": {"x": 40.0, "y": 0.0}}, {"lineTo": {"x": 12.0, "y": 28.0}}]}, "Point3D_P": {"type": "Point", "position": {"x": 0.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_P": {"type": "Label", "position": {"x": 0.0, "y": -5.0, "z": 20.0}, "text": "P", "position_offset": {"x": 0.0, "y": -4.8, "z": 0.0}}, "Point3D_A": {"type": "Point", "position": {"x": 40.0, "y": -5.0, "z": 20.0}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": 40.0, "y": -5.0, "z": 20.0}, "text": "A", "position_offset": {"x": 4.762933808225604, "y": -0.5953667260282005, "z": 0.0}}, "Point3D_N": {"type": "Point", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": 12.0, "y": -4.9999999998628955, "z": -8.0}, "text": "N", "position_offset": {"x": 1.8658470193695444, "y": -0.7774362580493255, "z": -4.353643045195604}}, "Point3D_T": {"type": "Point", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_T": {"type": "Label", "position": {"x": 0.0, "y": 35.0, "z": 20.000000000195865}, "text": "T", "position_offset": {"x": 0.0, "y": 4.8, "z": 2.6861438787558914e-11}}, "Point3D_C": {"type": "Point", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 40.0, "y": 35.0, "z": 20.000000000195865}, "text": "C", "position_offset": {"x": 3.612368134593013, "y": 3.160822117768887, "z": 1.7688381215585684e-11}}, "Point3D_O": {"type": "Point", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}}, "Label_for_Point3D_O": {"type": "Label", "position": {"x": 12.0, "y": 35.000000000137106, "z": -7.999999999804135}, "text": "O", "position_offset": {"x": 1.2413677786677848, "y": 3.6206560211285552, "z": -2.8965248168712363}}}}

On pourra faire pivoter la figure en cliquant dessus.

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 2 : Reconnaître un solide, ses sommets/faces/arêtes

On considère le solide ci-dessous :

{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240}, "size": {"width": 300, "height": 300}, "materials_options": {"surface": {"color": 16744448}, "solid_line": {"color": 16744448, "linewidth": 2}, "hidden_solid_line": {"color": 16744448, "linewidth": 2}}}, "data": {"Cuboid_(HIDVOUWR)": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 25, "ly": 10, "lz": 40, "point_render": {"visible": false}, "label_render": {"visible": true}}, "Label_for_Point3D_H": {"type": "Label", "position": {"x": -13.0, "y": -5.0, "z": -20.0}, "text": "H", "position_offset": {"x": -2.5603030123701207, "y": -0.9847319278346617, "z": -3.9389277113386467}}, "Label_for_Point3D_I": {"type": "Label", "position": {"x": 12.0, "y": -5.0, "z": -20.0}, "text": "I", "position_offset": {"x": 2.4147176149458347, "y": -1.0061323395607644, "z": -4.024529358243058}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": 12.0, "y": 5.0, "z": -20.0}, "text": "D", "position_offset": {"x": 2.4147176149458347, "y": 1.0061323395607644, "z": -4.024529358243058}}, "Label_for_Point3D_V": {"type": "Label", "position": {"x": -13.0, "y": 5.0, "z": -20.0}, "text": "V", "position_offset": {"x": -2.5603030123701207, "y": 0.9847319278346617, "z": -3.9389277113386467}}, "Label_for_Point3D_O": {"type": "Label", "position": {"x": -13.0, "y": -5.0, "z": 20.0}, "text": "O", "position_offset": {"x": -2.5603030123701207, "y": -0.9847319278346617, "z": 3.9389277113386467}}, "Label_for_Point3D_U": {"type": "Label", "position": {"x": 12.0, "y": -5.0, "z": 20.0}, "text": "U", "position_offset": {"x": 2.4147176149458347, "y": -1.0061323395607644, "z": 4.024529358243058}}, "Label_for_Point3D_W": {"type": "Label", "position": {"x": 12.0, "y": 5.0, "z": 20.0}, "text": "W", "position_offset": {"x": 2.4147176149458347, "y": 1.0061323395607644, "z": 4.024529358243058}}, "Label_for_Point3D_R": {"type": "Label", "position": {"x": -13.0, "y": 5.0, "z": 20.0}, "text": "R", "position_offset": {"x": -2.5603030123701207, "y": 0.9847319278346617, "z": 3.9389277113386467}}}}

On pourra faire pivoter la figure en cliquant dessus.

Quelle est sa nature ?

Combien a-t-il de sommets ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il de faces ?

Les nommer en les séparant par des points-virgule.

Combien a-t-il d'arêtes ?

Les nommer en les séparant par des points-virgule.

Exercice 3 : Construire un solide à partir d'un patron et le reconnaître

Voici le patron d'un solide tel que \( UR = 2 cm \).

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1. Construire ce solide
2. De quel type de solide s'agit-il ?

Exercice 4 : Se repérer sur une perspective cavalière

On s'intéresse à la représentation dans le plan d'un pavé droit ci dessous :

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Compléter les phrases suivantes avec ces mots ou expressions : parallèles, perpendiculaires, sécantes ou ni sécantes ni parallèles.
Pour deux droites sécantes et perpendiculaires, on notera "perpendiculaires"

Dans le plan, les droites \((AE)\) et \((BF)\) sont .

Dans le plan, les droites \((DH)\) et \((AE)\) sont .

Dans le plan, les droites \((BC)\) et \((CD)\) sont .

On s'intéresse maintenant au pavé droit, dans l'espace :

Dans l'espace, les droites \((AB)\) et \((EH)\) sont .

Dans l'espace, les droites \((GC)\) et \((AD)\) sont .

Dans l'espace, les droites \((EA)\) et \((FG)\) sont .

Exercice 5 : Calculer le volume d'un solide à partir d'une unité de volume

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0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_99": {"position": {"x": -18.0, "y": 0.0, "z": 9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_108": {"position": {"x": -18.0, "y": 9.0, "z": -18.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_117": {"position": {"x": -18.0, "y": 9.0, "z": -9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_126": {"position": {"x": -18.0, "y": 9.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_135": {"position": {"x": -18.0, "y": 9.0, "z": 9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_144": {"position": {"x": -9.0, "y": -18.0, 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9.0, "y": -9.0, "z": -9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_477": {"position": {"x": 9.0, "y": -9.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_486": {"position": {"x": 9.0, "y": 0.0, "z": -18.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_495": {"position": {"x": 9.0, "y": 0.0, "z": -9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_504": {"position": {"x": 9.0, "y": 9.0, "z": -18.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_513": {"position": {"x": 9.0, "y": 9.0, "z": -9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_522": {"position": {"x": 9.0, "y": 9.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}, "Cuboid_531": {"position": {"x": 9.0, "y": 9.0, "z": 9.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "Cuboid", "lx": 9, "ly": 9, "lz": 9}}}

De combien de cubes le solide ci-dessus est-il composé ?
Remarque : Penser à retourner le solide pour voir toutes ses faces.

Combien manque-t-il de petits cubes à ce solide pour former un grand cube ?

Revenir au choix du niveau